Subgroups — Q₈×Dic₇₀₆ #36378518

Order 22592.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014992	C₂	1	3	yes	maximal
2	#26014991	C₂	1	6	yes	maximal
3	Q₈×Dic₃₅₃	C₂	1	2	yes	maximal
4	C₄×Dic₇₀₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₀₆	C₂²	1	6	yes	—
7	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₅₃	C₂²	1	6	yes	—
9	C₄⋊Dic₃₅₃	C₂²	1	3	yes	—
10	#19150716	C₂²	1	6	yes	—
11	C₄×Dic₃₅₃	C₂²	1	6	yes	—
12	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₅₃	C₂²	1	3	yes	—
14	C₄²×C₃₅₃	C₂²	1	3	yes	—
15	Dic₇₀₆	—	4	6	no	—
16	Dic₇₀₆	Q₈	1	4	yes	direct product factor
17	C₂×Dic₃₅₃	C₂³	1	6	yes	—
18	C₂×Dic₃₅₃	C₂³	1	2	yes	—
19	Q₈×C₃₅₃	Q₈	1	4	yes	complemented
20	C₂×C₄×C₃₅₃	C₂³	1	3	yes	—
21	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	C₂³	1	3	yes	—
23	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
24	C₄×C₃₅₃	—	2	3	no	—
25	C₄×C₃₅₃	C₂×Q₈	1	2	yes	—
26	C₄×C₃₅₃	C₂×Q₈	1	6	yes	—
27	Dic₃₅₃	—	2	6	no	—
28	Dic₃₅₃	C₂×Q₈	1	4	yes	—
29	C₂×C₃₅₃	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₃₅₃	C₂²×Q₈	1	1	yes	characteristic
31	C₂×C₃₅₃	H(5,𝔽₂)	1	1	yes	characteristic
32	C₃₅₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	353	1	no	maximal, Sylow
34	C₄⋊Q₈	—	353	6	no	—
35	C₄⋊Q₈	—	353	3	no	—
36	C₄×Q₈	—	353	2	no	—
37	C₄×Q₈	—	353	3	no	—
38	C₄×Q₈	D₃₅₃	1	1	yes	characteristic
39	C₂×Q₈	—	353	1	no	—
40	C₂×Q₈	—	353	6	no	—
41	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
42	C₄⋊C₄	—	353	6	no	—
43	C₄⋊C₄	—	353	3	no	—
44	C₄⋊C₄	—	353	6	no	—
45	C₄⋊C₄	C₂×D₃₅₃	1	3	yes	—
46	C₄²	—	353	6	no	—
47	C₄²	C₂×D₃₅₃	1	3	yes	—
48	Q₈	—	353	4	no	—
49	Q₈	—	1412	6	no	—
50	Q₈	Dic₇₀₆	1	4	yes	direct product factor
51	C₂×C₄	—	353	2	no	—
52	C₂×C₄	—	353	6	no	—
53	C₂×C₄	C₂²×D₃₅₃	1	3	yes	—
54	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₃₅₃	1	3	yes	—
56	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	353	4	no	—
59	C₄	—	706	6	no	—
60	C₄	Q₈×D₃₅₃	1	2	yes	—
61	C₄	C₂×Dic₇₀₆	1	6	yes	—
62	C₂	C₂×Q₈×D₃₅₃	1	1	yes	characteristic
63	C₂	C₂²×Dic₇₀₆	1	1	yes	characteristic
64	C₂	#26015038	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₃
commutator, socle, prime radical, cosocle	C₂²×C₃₅₃	C₂⁴
Fitting	C₄×Q₈×C₃₅₃	C₂
hypercenter	C₄×Q₈	D₃₅₃
hypocenter	C₃₅₃	Q₈²
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
16. Dic₇₀₆	Q₈	direct (N × H)
19. Q₈×C₃₅₃	Q₈	semidirect
32. C₃₅₃	Q₈²	semidirect
50. Q₈	Dic₇₀₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26014992	2
#26014991	2
Q₈×Dic₃₅₃	2
C₄×Dic₇₀₆	2
C₄×Q₈×C₃₅₃	2
Q₈²	353

Subgroups — Q8×Dic706 #36378518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₇₀₆ #36378518